A Subtle Symmetry of Lebesgue's Measure

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Uludag M., Ayral H.

JOURNAL OF THEORETICAL PROBABILITY, vol.32, no.1, pp.527-540, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 32 Issue: 1
  • Publication Date: 2019
  • Doi Number: 10.1007/s10959-017-0804-y
  • Journal Name: JOURNAL OF THEORETICAL PROBABILITY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.527-540
  • Keywords: Farey tree, Modular group, Boundary, Measure, Lebesgue measure, Stern-Brocot tree, Calkin-Wilf tree, FUNCTION EXPLICIT SERIES, DYADIC PERIOD FUNCTION
  • Galatasaray University Affiliated: Yes

Abstract

We represent the Lebesgue measure on the unit interval as a boundary measure of the Farey tree and show that this representation has a certain symmetry related to the tree automorphism induced by Dyer's outer automorphism of the group PGL(2,Z). Our approach gives rise to three new measures on the unit interval which are possibly of arithmetic significance.